In the U.S., you fill in the blanks. If there are blank spaces for you to write answers, you fill them in. If you answer a question by coloring in the circle of the right answer, you fill it in. For those, you could not fill out. If you have an entire form to answer questions on, though, a whole page that needs to be answered (even by filling in the blanks!), you can say you filled out the form, although filled in would still work too.

Fill out can also mean take up all of the space or make something better and more varied. If there were a program that you were putting together, and you decided to take up the remaining time you had with a poem you read, for example, someone could say, "I thought that poem you read filled out the program quite nicely." A growing child can fill out his clothes, perhaps, and a side dish can fill out a meal.

There is no difference in American english, really. We say both. I suppose fill in would be used to fill in a hole, or a form with bubbles. You fill out a form where things are long and need to be written out.